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Five-fold Symmetry in AI!
By!
Ian Beardsley!
Copyright © 2021 by Ian Beardsley"
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In four parts we examine artificial intelligence and biological life, their connection with one
another and suggest this connection might have clues that can help us find the origin of life. !
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Part 1!
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To make artificial intelligence (AI) we need semiconductors, like diodes and transistors. To
make semi conductors we need to dope Silicon Si 4- with a group 13 doping agent to have
positive silicon such as with boron B 3- or with a group 15 doping agent like phosphorus P 5-
to have negative type silicon. Or we can dope germanium Ge 4- with a group 13 doping agent
like gallium Ga 3- for positive type germanium or with a group 15 doping agent like arsenic As
5- to have negative type silicon. We connect the negative with the positive to have a
semiconductor, meaning a current can run through it in only one direction. !
We pull these AI elements out of the periodic table of the elements to make an AI periodic
table:!
We now notice we can make a 3 by 3 matrix of it, which lends itself to to the curl of a vector
field, by including biological elements carbon C (above Si):!
=!
=!
=!
!
Let us dot this with and take the double integral over Si to Ge
over both variable sets:!
i
j
k
∂
∂x
∂
∂y
∂
∂z
(C ⋅ P)y (Si ⋅ G a)z (Ge ⋅ As)y
(Ge ⋅ As − Si ⋅ G a)
i + (C ⋅ P)
k
[
(72.64)(74.92) − (28.09)(69.72)
]
i +
[
(12.01)(30.97)
]
k
3,482
(
g
mol
)
2
i + 372
(
g
mol
)
2
k
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=!
=!
=!
=!
!
!
Now let us take the harmonic mean between Si and Ge. It is!
!
And the arithmetic mean between them:!
!
We see the value of 44.3 g/mol is somewhere between the harmonic and arithmetic mean.
Perhaps it is the geometric mean…!
!
Thus we can say…!
∫
Ge
Si
∫
Ge
Si
(
3,483
(
g
mol
)
2
i + 372
(
g
mol
)
2
k
)
⋅
(
zd ydz
i + yd xd y
k
)
∫
Ge
Si
∫
Ge
Si
(
3,483
(
g
mol
)
2
⋅ zdzd y + 372
(
g
mol
)
2
⋅ yd xd y
)
∫
Ge
Si
3,483
(
(72.64 − 28.09)
2
2
)
dy +
∫
Ge
Si
372y ⋅ (72.64 − 28.09)d y
3456359
(
g
mol
)
4
(72.64 − 28.09) + 16573
(
g
mol
)
3
(
(72.64 − 28.09)
2
2
)
170427030.8
(
g
mol
)
5
5
170427030.8 = 44.3g/mol
2SiGe
Si + Ge
= 40.5g/mol
Si + Ge
2
= 50.365g/m ol
SiGe = 45g/mol
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!
!
Which like Stoke’s Theorem in that it relates an integral of a flux over a surface to path integral.
The expression on the right-hand side of the equation is the geometric mean between Si and
Ge. This integral can better be represented with product calculus:!
!
Where and and n=2. If we we say the arithmetic mean is A, and the harmonic
mean is H, the geometric mean G…!
!
This is!
!
This is quite interesting because!
!
!
!
I say interesting because we can write all three of these as one equation, the f-mean:!
!
The harmonic mean and the arithmetic mean are special cases of the power-mean which is the
case when , the harmonic mean when p=-1, and the arithmetic mean when p=1.!
u = (CP ⋅ y, SiG a ⋅ z, G a As ⋅ y)
5
∫
Ge
Si
∫
Ge
Si
∇ ×
u ⋅ d
a = exp
(
1
Ge − Si
∫
Ge
Si
ln(x)d x
)
5
∫
Ge
Si
∫
Ge
Si
∇ ×
u ⋅ d
a =
n
n
∏
i=1
x
i
x
1
= Si
x
2
= Ge
A + H
2
= 45.4325 ≈ G
Si
2
+ 6SiGe + Ge
2
4(Si + Ge)
≈ SiGe
H(a, b) =
1
1
b − a
∫
b
a
dx
x
A(a, b) =
1
b − a
∫
b
a
xd x
G(a, b) = exp
(
1
Ge − Si
∫
b
a
ln(x)d x
)
M
f
(x
1
, …x
n
) = f
−1
(
1
n
n
∑
i=1
f (x
i
)
)
f (x) = x
p
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But what is interesting to me is that to get the geometric mean from the f-mean we have to
change the function f(x) to f(x)=ln(x). This is when it becomes simpler to express the geometric
mean in terms of product notation:!
!
And this is precisely interesting to me because five-fold geometry does a similar thing. We have
a five-fold expression in our AI equation we arrived at:!
!
In that we take the fifth root of the double integral on left. This makes me think of how we can
tile a surface with regular polygons the 3-sided (triangle), 4 sided (square), and 6-sided (regular
hexagon) but five pops out and the pentagon requires another shape added in to tile a surface
without leaving gaps as a so-called Archimedean tessellator, the equilateral triangle, square,
and regular hexagon are the regular tessellators. However, if you are working with solids, there
are five regular solids and they all tile to close off a space, using triangles, for example the
tetrahedron, or squares (the cube), and yes the regular pentagon in the dodecahedron.!
See illustration on next page…"
M
0
(x
1
, . . x
n
) =
n
n
∏
i=1
x
i
5
∫
Ge
Si
∫
Ge
Si
∇ ×
u ⋅ d
a =
n
n
∏
i=1
x
i
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"
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It was the Russian scientist Shubnikov who noticed that five-fold symmetry is more
characteristic of life while six-fold symmetry is more characteristic of the physical. He wrote:!
As to the alive organisms, we have not for them theory, which could answer the question what
kinds of symmetry are compatible or incompatible to existence of living material. But we can
note here that remarkable fact that among the representations of the alive nature the
pentagonal symmetry meets more often.
I think from experience and observation you will find this as true if you pay close attention to
Nature. You will find if you look at flowers every now and then you will find six petals around its
center, or sometimes as with a rose perhaps near a hundred petals, but most often you will find
there are five petals around the center of a flower. As well, even in the rose, with near a
hundred petals, they spiral in as a golden spiral, which is built of ratios of the golden ratio (
and use patterns of Fibonacci numbers. The successive ratios between terms in the Fibonacci
sequence converge on at infinity and the golden ratio is derived from pentagonal symmetry
in that if you draw in the chord of a regular pentagon, the ratio of it to its side is . And indeed
the human has two legs, two arms and a head adding up to five, or two eyes, and a nose and a
mouth adding up to five. Or, five fingers, or five toes on each hand or each foot. But for the
physical like a snowflake, there are six points that form around it giving it hexagonal symmetry.
The starfish has five arms.!
In looking at life we notice it is based on carbon which is in group 14 of the periodic table of
the elements just like semiconductor elements silicon and germanium. It is because of this that
carbon works because it means has 4 valence electrons, meaning it can form long chains with
hydrogen making organic matter the hydrocarbons, utilizing oxygen (O), nitrogen (N),
phosphorus (P), and sulfur (S). Life does not seem to be based on silicon, though, even though
it has 4 valence electrons as well because while carbon can combine with hydrogen to make
hydrocarbons such as CH4, or combine with O, N, H to make the most simple organic
compound isocyanic acid HNCO which binds H-N=C=O, silicon in the presence of oxygen
forms glass SiO2 so easily that it can not combine with the H, N, C, O, P, and S readily with
each equally so as to form functional hydrocarbons.!
It is at this point that I would like to note that carbon is element six in the periodic table giving it
6 protons, and since its molar mass is 12.01, it has 6 neutrons. It so happens that closest
packing of equal radius spheres in the plane like protons, and neutrons is six-around one or
hexagonal symmetry. As Buckminster Fuller constructed his geometry in Synergetics, he
outlined his discovery that equal-radius spheres pack in the form of what he called the vector
equilibrium, which is the cuboctahedron, which he demonstrated was the most transformable
construct and as such becomes pivotal to his Synergetics,!
I would like to suggest in light of this that since carbon has six protons and six electrons, with
the six protons determining its number of electrons (6 to be neutral) giving it four valence
electrons in its outer shell for combining with other elements (the outer shell is four and wants
four to complete an octet, such as four hydrogens each H+, that though life more often meets
with pentagonal symmetry, and here we see carbon meets with six-around-one in the plane, or
twelve-around-one in space as the vector equilibrium, or six-fold symmetry, it is because life is
built out of the physical, like carbon to make the biological, characteristic of pentagonal
symmetry. And it is here I suggest that life animates out of a dynamic structuring of the
physical (inanimate). See illustration on the next page…"
Φ)
Φ
Φ
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"
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Indeed we see life could be the interplay between 3, 4, 5, 6 as structured in Buckminster
Fuller’s Synergetics. For instance the vector equilibrium (cuboctahedron) is made of equilateral
triangles and squares, the regular tessellators. With eight triangles and six squares. All of this
speaks respectively of NH3 (ammonia, believed to have contributed to making the amino acids
the building blocks of life) which is three hydrogens around a Nitrogen, CH4 (methane, believed
to have contributed to the formation of amino acids in primordial earth as well) the eight
triangles in the cuboctahedron representing the combination of elements such that they
complete an octet, and its six squares, the six protons, six neutrons, and six electrons of
carbon.!
With all said here so far, it might be said that understanding life and its origins can be
understood by looking at artificial intelligence."
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Part 2!
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Let us return to the geometric mean becoming a different function in the f-mean. We have:!
!
!
p=1 yields:!
!
Is the arithmetic mean between x1 and x2. Now take p=-1:!
=!
!
Is the harmonic mean between x1 and x2. Now we try p=0 hoping to get the geometric mean…!
=!
=!
!
So for we can’t make sense and we have to search for a function that will produce
the geometric mean in the f-mean. It is ln(x). This is interesting because the natural log of x was
created to settle the following conundrum:!
!
M
f
= f
−1
(
1
n
n
∑
i=1
f (x
i
)
)
M
f
(x
1
, x
2
) = f
−1
(
1
n
2
∑
i=1
x
p
i
)
p
M
f
(x
1
, x
2
) =
(
1
2
x
1
+
1
2
x
2
)
=
x
1
+ x
2
2
M
f
(x
1
, x
2
) =
(
1
2
2
∑
i=1
x
−1
i
)
−1
2
1
x
1
+
1
x
2
=
2x
1
x
2
x
1
+ x
2
M
f
(x
1
, x
2
) =
(
1
2
2
∑
i=1
x
i
0
)
1
0
(
1
2
2
∑
i=1
1
)
1
0
(
1
2
)
∞
f (x) = x
p
∫
d x
x
=
∫
x
−1
d x =
x
−1+1
0
=
x
0
0
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This is where we need to create the natural logarithm function so we can have a solution to
such an integral and, we have!
!
Where!
!
!
Let us return to our . It is not a sum!
!
But is a product!
=!
!
What this says is that what is important is not the values of data points in an experiment, not
the ’s but the i’s themselves, the number of the data point. Like the one in measurement 1,
the 2 in measurement 2. Never mind that measurement 1 might equal 2.3 grams, measurement
3 might equal 0.5 grams, the important thing is the 1/2 outside the parenthesis because we are
taking, which is is always 1. This is how reality never has any meaning: we
just change to f(x)= ln(x), which is the equivalent of writing!
!
Thus it is the experience itself that counts, we find !
!
If we say e=2.718…"
∫
d x
x
= ln(x) + C
ln(x) = log
e
(x)
e = 2.718…
(
1
2
)
∞
1
2
∞
∑
i=1
i
i
=
1
2
(
1
1
+
2
2
+
3
3
+ …
)
M
0
(x
1
, x
2
, x
3
…x
n
) =
1
2
∞
n
∏
i=1
i
i
1
2
(
1
1
⋅
2
2
⋅
3
3
⋅ …
)
x
i
i /i
1/1,2/2,3/3,...
f (x) = x
p
G =
n
n
∏
i=1
x
i
∫
d x
x
= ln(x) + C
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Part 3!
of 16 27
!
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!
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!
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!
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Part 4!
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As we can account for some of the amino acids, the building blocks of life, in that Miller and
Urey showed eleven of the twenty amino acids in proteins can be made by mixing together the
primordial earth substances CH4 (methane) NH3 (Ammonia) and H2O water with a few other
gases that were present, and then applying electricity to simulate electrical storms, we need
not just amino acids to have life but sugars for the DNA and RNA that encode life.!
Sugars are carbohydrates with the formula:!
!
Where n is 2 to 7. n=1 is not a sugar and is called formaldehyde which is:!
!
Which has the same structure as a sugar, which is a monomer from which the sugars form; that
is, sugars are polymers of formaldehyde. For DNA (deoxyribose nucleic acid) and RNA (ribose
nucleic acid) that encode life we need deoxyribose:!
!
And, ribose:!
!
The sugar produced by plants through photosynthesis that serves for its food is glucose
which is n=6 in . To make these sugars formaldehyde first combines to
make the sugar glyceraldehyde (n=3):!
!
Then combines with this. But to explain the origins of life in terms of arising from a primordial
substance, we need to explain how we make formaldehyde.!
Formaldehyde is an intermediate in the combustion or oxidation of methane (one of the
primordial gases that make some of the amino acids). It does not accumulate in the
environment because it is broken down by sunlight or by bacteria in soil and water. It is
produced by the action of sunlight and oxygen on atmospheric methane. In the lab it is stored
as an aqueous solution (formalin) because it polymerizes spontaneously into
paraformaldehyde. It exist in the interstellar medium (the empty space between stars) and is
proposed to be formed there by the hydrogenation of carbon monoxide ice:!
!
!
C
n
(H
2
O)
n
CH
2
O
C
5
H
10
O
4
C
5
H
10
O
5
C
6
H
12
O
6
C
n
(H
2
O)
n
C
3
H
6
O
3
H + CO ⟶ HCO
HCO + H ⟶ CH
2
O
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In order to determine whether life can arise spontaneously or not, a brief review of what we
know suggests what I call an activation function.!
Miller Urey Chemistry!
Under Nobel prize winner in chemistry (1934) Harold Urey, Stanley Miller a graduate student set
out to see if he mimicked the theoretical primordial Earth, he could produce the 20 amino acids
that are the building block of life. He created the ocean by filling a closed glass container with
water, and coming out of this was a tube that went to a second chamber that mimicked the
primordial earth atmosphere which was methane, ammonia, and hydrogen. As the heated
water vaporized it flowed out of the first chamber and into the second chamber. He passed
electricity through the second chamber to mimic electrical storms or lightning. Between the
first chamber and second he placed a condenser, so when the water vapor rose into it, some of
it would condense into liquid to mimic rain. With this experiment he produced 10 of the 20
biological amino acids. We have yet to find a way to produce all 20 under theoretical primordial
earth conditions.!
Produced!
Glycine, Alanine, Aspartate, Valine, Leucine, Glutamate, Isoleucine, Serine, Proline, Threonine!
Not Produced!
Phenyalanine, Tyrosine, Tryptophan, Histidine, Lysine, Arginine, Cysteine, Methionine,
Asparagine, Glutamine!
H2O!
Water is an extraordinary substance and in order to have life you need it. The earth is not only
the right distance from the sun for water to exist in three phases (ice, vapor, and liquid) but is
also happens to be very plentiful here, it covers three quarters of the planet’s surface and what
is more there are great amounts of it under its surface. Let’s look at some of the properties of
water that it has that allows for life:!
1. Water is solvent meaning it dissolves a great number of substances.!
2. Water is cohesive and adhesive, cohesive because it flows freely, yet adhesive in that can
also adhere to surfaces. Unicellular organisms rely on external water to transport nutrients
and waste while multicellular organisms have internal vessels that use it to do the same.
Because of adhesion and cohesion water can climb up from the roots of a tree to its top by
tension created by water evaporating from its leaves.!
3. Water has a high surface tension meaning plant debris can rest on its surface providing
food and shelter for aquatic life.!
4. Water in its solid phase (ice) is less dense than it is in its liquid phase because when it
freezes it expands meaning it floats on the surface water. If it was not for this life could not
exist on earth because if the ice sank the ponds, lakes, and perhaps even the oceans
would freeze over solid.!
5. Water has a high heat capacity. The specific heat of water is one calorie per gram degree
centigrade which means it takes one calorie to raise the temperature of a gram of it by one
degree centigrade. This keeps the earth relatively cool, and thus life thrives. A lot of the
sunlight’s energy goes into vaporizing it into clouds that would otherwise go into heating
the planet.!
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Dehydration Synthesis and Hydrolysis!
Let us look at how water synthesizes substances and breaks them down. Take making fat from
glycerol and a fatty acid:!
!
If you heat it you remove an H from the glycerol and an OH (hydroxide) from the fatty acid
which is to remove a water molecule H2O leaving an O in the glycerol and a C in the fatty acid
that joins the glycerol with the fatty acid to make the fat. For hydrolysis you add an OH to the
fatty acid and and H to the glycerol by adding water (H2O) to the fat thus breaking down it
down into a fatty acid and glycerol.!
DNA and RNA!
Life is encoded by DNA (deoxyribose nucleic acid) and RNA (ribonucleic acid) which make
nucleic acids. Each nucleic acid is a monomer in a polymer called a nucleotide. The monomer
consists of a phosphate, a 5 carbon sugar, and a nitrogen containing base. The phosphate is
the phosphate ion . Deoxyribose is the sugar and is in DNA and ribose is the
sugar . The bases are guanine, cytosine, adenine, thymine in DNA and RNA uses
uracil in place of thymine. The phosphate combined with the sugar is the backbone of DNA
and RNA, and the bases are attached to the backbone. There are two back bones running
parallel to one another and the bases of one attach to the bases of the other in a pairing that is
always guanine (G) pairs with cytosine (C) and adenine (A) pairs with thymine (T). It is the
sequencing of these pairings the encodes for life, and the parallel backbones are twisted so
you have a twisted ladder where the base pairings are its rungs. !
Guanine: !
Cytosine: !
Adenine: !
Thymine: , Uracil: !
PO
4
−
C
5
H
10
O
4
C
5
H
10
O
5
C
5
H
5
N
5
O
C
4
H
5
N
3
O
C
5
H
5
N
5
C
6
H
6
N
2
O
2
C
4
H
4
N
2
O
2
of 24 27
"
of 25 27
The Problem!
In order to have life we need to have the nucleic acids, which means we need to have the
bases guanine, cytosine, adenine, thymine, and uracil. Can they arise spontaneously from the
calculated conditions of the primordial earth?!
John Oro in 1961 found amino acids and adenine could form from the mixture of hydrogen
cyanide and ammonia in water. Later researchers found several of the bases needed were
present if they allowed hydrogen cyanide to combine with the ammonia when heated in acid.
The problem is hydrogen cyanide present in the laboratory experiment was hundreds of
thousands of times more concentrated than is calculated to have existed on the primordial
earth surface. Further, hydrogen cyanide cannot be concentrated by the evaporation of sea
water in a tidal pool because it is more volatile than water,.!
Later, Leslie Orgel found that freezing a hydrogen cyanide solution would allow it to form in the
voids between ice crystals, which meant adenine could only form in the frozen polar regions. In
1975 Miller froze the stuff for 27 years, then analyzed it finding small amounts of several of the
bases including the adenine.!
We now know that four molecules of hydrogen cyanide can combine to form
diaminomaleonitrile, then, under sunlight, if it reacts with another molecule of hydrogen
cyanide it produces adenine in 7% yield. But if four molecules react with salt ammonium
formate there is 90% yield of adenine. However, this requires dehydration, by removing two
molecules of water, meaning we need to boil away the water of the solution to dryness.!
The Solution!
Substances and rapid temperature changes (cold for some reactions, warm for others like
dehydration synthesis) that are not present today, nor that we calculate were present in the
primordial earth (like sufficient quantities of hydrogen cyanide) that are needed to account for
a prebiotic pathway to the nucleobases and their combination with phosphates and sugars,
and all of the biological amino acids would suggest there was the presence of what I will call an
activation function. Since these necessary substances are not present today, and cannot be
calculated to have existed a long time ago, I suggest that the activation function was a limiting
factor, that as it activated life from what was present, it depleted determining how much life
was present in the beginning by its total depletion. Once life exists the production of the
nucleic acids is possible because it can now be powered by life’s consumption of
carbohydrates, which only can exist on the earth after the existence of life."
of 26 27
The Activation Function!
I would like to suggest that the prebiotic chemistry might have been passed through an
activation function that disappeared after life was on its way to evolving. !
The problem, then, of answering the question of how life began is one of finding the activation
function and its mechanism by which it takes prebiotic chemistry and activates it (makes it
alive) so it can now self-replicate, and evolve. We assume that as this mechanism activates the
molecules, its mechanism depletes as it activates from what is available. In this sense the
mechanism is a limiting reactant, so it determines how much material is activated before it
depletes completely.!
Logically, the way to determine what this mechanism is, and how it serves as an activation
function is to look for the by-products of the reaction that are left over, and from that, deduce
its nature.!
To do this, we have to look for that thing in our knowledge of the Earth’s history that does not
make sense. This would be in the faint young star paradox. We know that five billion years ago,
when the Earth and Sun first formed, that the sun was 0.7 times its present output and so, the
Earth should have been frozen over, yet, we know it was not. That it had water in its liquid
phase. Thus something was there that is not present today. That something must have been
the mechanism for the activation function that “turned on” prebiotic chemistry.!
If is the activation function, where x is the prebiotic material, and we say is residue of
the reaction, and is the activated substance (life) then,!
!
We know . If we can find in nature, we can deduce .!
I have presented it like this because 1) Life has not been created in the laboratory from scratch
2) New life does not seem to be originating on earth in present times. Therefore, the activation
function is probably not present on Earth today and more than likely disappeared, or depleted
after activating prebiotic chemistry. Life exists, yet we do not know how prebiotic substances
organize into self-replicating systems that evolve. Therefore, we must look for something
concerning the Earth that does not make sense. I suggest that would be the young star
paradox. If the Earth had water in its liquid phase when it should have been frozen over, then
something could have existed then that was a limiting reactant, or something like it, that
activated prebiotic substances, in that it was responsible for warming the earth (perhaps a heat
retaining substance).!
I use the term limiting reactant loosely as well as prebiotic chemistry because one, the reactant
was not necessarily a substance alone, but a manifestation of energy not just necessarily
sunlight incident upon the earth and, the prebiotic chemistry was not necessarily just
substances that existed then from what we have theoretically calculated."
σ (x)
r
l
l + r = σ (x)
l
r
σ (x)
of 27 27
The Author!
!